Target Function without Local Minimum for Systems of Logical Equations with a Unique Solution

Abstract

Many of the applied algorithms that have been developed for solving a system of logical equations or the Boolean satisfiability problem have solved the problem in the Boolean domain. However, other approaches have recently been developed and improved. One of these developments is the transformation of a system of logical equations to a real continuous domain. The essence of this development is that a system of logical equations is transformed into a system in a real domain and the solution is sought in a real continuous domain. A real continuous domain is a richer domain, as it involves many well-developed algorithms. In this paper, we have constructively transformed the solution of any system of logical equations with a unique solution into an optimization problem for a polylinear target function in a unit n-dimensional cube Kn. The resulting polylinear target function in Kn does not have a local minimum. We proved that only once by calculating the gradient of the polylinear target function at any interior point of the Kn cube, we can determine the solution to the system of logical equations.

Keywords

• polylinear functions
• global optimization
• algorithms
• Zhegalkin polynomials
• logical operations
• systems of Boolean algebraic equations